Let \(\Theta _j:=\Lambda _j\pi _{W_j}S_{\Lambda }^{-\frac{1}{2}}\)  and \(X_j:=S_{\Lambda }^{-\frac{1}{2}}W_j\) . In [1]} has been showed that \((X_j, \Theta _j, v_j)\)  is a Parseval g-fusion frame for \(H\) . Then, by Corollary REF , we have

\(\sum _{j\in I}v_j^2\Vert \Theta _j \pi _{X_j}f\Vert ^2-\Vert \sum _{j\in I}v_j^2 \pi _{X_j}\Theta ^*_j &\Theta _j \pi _{X_j}f\Vert ^2 =\\&=\sum _{j\in I^c}v_j^2\Vert \Theta _j \pi _{X_j}f\Vert ^2-\Vert \sum _{j\in I^c}v_j^2 \pi _{X_j}\Theta ^*_j \Theta _j \pi _{X_j}f\Vert ^2.\) 


By replacing \(S_{\Lambda }^{\frac{1}{2}}f\)  instead of \(f\)  and this fact that
\(\sum _{j\in I}v_j^2 \pi _{X_j}\Theta ^*_j \Theta _j \pi _{X_j}f&=\sum _{j\in I}v_j^2 (\Theta ^*_j\pi _{X_j})^* \Theta _j \pi _{X_j}f\\&=\sum _{j\in I}v_j^2(\Lambda _j \pi _{W_j}S_{\Lambda }^{-\frac{1}{2}}\pi _{X_j})^*\Lambda _j \pi _{W_j}S_{\Lambda }^{-\frac{1}{2}}\pi _{X_j}f\\&=\sum _{j\in I}v_j^2 S_{\Lambda }^{-\frac{1}{2}}\pi _{W_j}\Lambda _j^*\Lambda _j \pi _{W_j}S_{\Lambda }^{-\frac{1}{2}}f\\&=S_{\Lambda }^{-\frac{1}{2}}S_{I}S_{\Lambda }^{-\frac{1}{2}}f,\) 

the proof follows.
